Transcript Ffwythiant Dwysedd Tebygolrwydd f(x)

Ffwythiant Dwysedd Tebygolrwydd f(x)
Probability Density Function f(x)
Os oes gan hapnewidyn di-dor werth posib rhwng a a b,
gallwn ddarlunio sut y rhennir un uned o debygolrwydd
rhwng y gwerthoedd yma mewn graff o f(x).
If a continuous random variable has a value between a and b, we can
show how one unit of probability is distributed in a graph of f(x).
f(x)
=1
a
b
x
Gan fod rhaid i gyfanswm yr arwynebedd o dan y gromlin
fod yn hafal i 1 cyfan (cyfanswm tebygolrwydd), mae
The area under the curve (the total probability) must be equal to 1,
therefore
b
 f ( x)dx  1
a
I ddarganfod P(c ≤ x ≤ d)
To calculate P(c ≤ x ≤ d)
f(x)
d

a
c d
b
x
f ( x ) dx
c
Pan fo X yn ddi-dor, gellir newid y symbol ≤ a rhoi < yn ei le
fel bod
P(c ≤ X ≤ d) = P(c < X < d) = P(c ≤ X < d) = P(c < X ≤ d)
When X is continuous, the ≤ symbol can be replaced with < so that
P(c ≤ X ≤ d) = P(c < X < d) = P(c ≤ X < d) = P(c < X ≤ d)
Enghraifft - Example
Dosrennir yr hapnewidyn di-dor X gyda ffwythiant dwysedd
tebygolrwydd f a roddir gan
X is a continuous random variable with a probability density function
f(x) = kx(4-x)
Darganfyddwch werth
Find the value of
a) k
b) P(X ≤ 3)
c) P(0 < X < 1 | X ≤ 3)
ar gyfer/for 0 ≤ x ≤ 4
4
a)
 kx(4  x)dx  1
0
4
k  ( 4 x  x 2 ) dx  1
0
4
 4x
x 
k
  1
3 0
 2
2
3
3
3




4
0
2
2
k  2(4)     2(0)    1
3 
3 

64

k 32    1
3

32 k
1
3
3
k
32
3
b) P(X ≤ 3) =
3
0 32 x(4  x)dx 
3
3
2
(
4
x

x
)dx 

32 0
3
3  2 x 
2 x   
32 
3 0
3
3
3



3 
3
0
2
2
 2(3)     2(0)   
32 
3 
3 
3
3 9
18  9  0 
32
32
27

32
c) P(0 < X < 1 | X ≤ 3) = P((0 < X < 1)  (X ≤ 3))
P(X ≤ 3)
P(A|B) = P(A  B)
P(B)
P((0 < X < 1)  (X ≤ 3)) = P(0 < X < 1)
1
P(0 < X < 1) =
3
2
(
4
x

x
)dx 

32 0
1
3 
2 1 3

2
x

x
32 
3 


0
3 
1 3 
1 3 
2
2
 2(1)  (1)    2(0)  (0)  

32 
3
3
 

3 5 5



32  3  32
5 27
P(0 < X < 1 | X ≤ 3) =

32 32
5

27
Ymarfer/Exercise 1.1
Mathemateg - Ystadegaeth Uned S2 – CBAC
Mathematics Statistics Unit S2 - WJEC
Gwaith Cartref/Homework 10